In this paper, we focus on Gaussian mixture families in which \(q_d(\theta ;\eta _d)\)  is the density of \(p\) -dimensional Gaussian \(N(\mu _d,\Sigma _d)\)  with \(\eta _d = (\mu _d,\Sigma _d)\) . For this case, [1]} showed that the infimum for () is reached when

dt+1=E[f(d ,t,t)]/dt+1,
